114 XIII INTEGRATION (1 3.5.2.1 ) n*(/) n n->oo Suppose first of all that/e <2fR(X) and that/, e jfR(X) for all n. It is clear that the supports of all the fn are contained in the same compact set K = Supp(/) u Supp(/i). By Dini's theorem (7.2.2), the sequence (/) con- verges uniformly to /on X. The formula (13.5.2.1) then follows from the fact that the restriction of n to Jf(X;K) is a continuous linear form on this Banach space. Now pass to the general case. Clearly we have ju*(/J ^ /**(/) for all n. Hence it is enough to show that, for each function u e jf R(X) such that u ^/ we have ju*(w) ^ sup /**(/). Now by (12.7.8) we know that, for each function/,, there exists an increasing sequence (gmn)m^i of functions belong- ing to 3ffR(X) such that / = sup gmn . We have /= sup gmn, hence also m m, n /= sup hn , where hn = sup #M . The functions hn clearly belong to *?f R(X) n P^n,q^n and form an increasing sequence. Since u ^/, we have u = sup(inf(w, An)); the n sequence of functions inf(w, hn) is increasing and belongs to Jf R(X); since also u eĞ2fR(X), the first part of the proof shows that /i*(w) = sup ju*(inf(w, /*)); n but since hn ^/, we have /^*(inf(w, /*)) ^ M*(/M), and therefore finally Q.E.D. We remark that because the functions in </ never take the value oo, the sum/ +/2 of two such functions is defined at every point of X, and belongs to J (12.7.5). (13.5.3) 7//i,/2 e J, then ^(f, +/2) = /.*(/) + We may write / = lim gn and /2 == lim ^w , where (gn) and (/ZM) are two increasing sequences of functions belonging to ^TR(X) (12.7.8). Hence we have / +/2 = lim (gn -f /?) (4.1.8). Since ]n(gn + hn) = /*(#) -f ju(^i)> the result follows from (13.5.2) and (4.1.8). Let (tn) be any sequence whose terms are real numbers ^0, or +00. Since the partial sums $ = /t + -f tn are defined (4.1.8) and form an ___ 00 increasing sequence, this sequence has a limit in R (4.2.1), denoted by J] tn and called the sum of the series with general term tn. For every sequence 00 (/,) of positive functions belonging to I, the function x\> ^fn(x) is therefore n = locally compact spaces and TT : X -> Y a proper continuous mapping